Several recent groundbreaking results in geometric measure
theory, homogeneous dynamics and number theory ultimately rely on a
key result of Bourgain known as Bourgain's Projection Theorem (of
course, each of these results require many other tools...
We use the min-max construction to find closed hypersurfaces
which are stationary with respect to anisotropic elliptic
integrands in any closed n-dimensional manifold . These surfaces
are regular outside a closed set of zero n-3 dimension. The...
In the early 80s Hatcher proved the Smale Conjecture, asserting
that the diffeomorphism group of the three-sphere retracts onto its
isometry group. The corresponding problem for RP^3 was open
nearly 40 years, and resolved only in 2019 by a detailed...
I will present recent work with Hairer, Rosati and Yi
establishing quantitative lower bounds for the top Lyapunov
exponent of linear PDEs driven by two-dimensional stochastic
Navier-Stokes equations on the torus. For both the
advection-diffusion...
Parallels between elliptic and parabolic theory of partial
differential equations have long been explored. In particular,
since elliptic theory can be seen as a steady-state version of
parabolic theory, if a parabolic estimate holds, then by...
In this talk, I will discuss some results concerning the
geometry and topology of manifolds on which the first eigenvalue of
the operator -γΔ + Ric is bounded below. Here, γ is a positive
number, Δ is the Laplacian, and Ric denotes the pointwise...
A minimal partition is a decomposition of a manifold into
disjoint sets that minimizes a spectral energy functional. In the
bipartite case minimal partitions are closely related to
eigenfunctions of the Laplacian, but in the non-bipartite case
they...
More than 120 years ago, Minkowski published a seminal paper
that laid the foundation for the field of convex geometry (as well
as several other areas of mathematics). Despite numerous advances
in the intervening years, there are fundamental...
Suppose f is a function with Fourier transform supported on the
unit sphere in Rd. Elias Stein conjectured in the 1960s that the Lp
norm of f is bounded by the Lp norm of its Fourier transform, for
any p>2d/(d−1). We propose to study this...
This talk is about the study of the Boltzmann equation in the
diffusive limit in a channel domain 𝕋2×(−1,1) nearby the 3D kinetic
Couette flow. We will begin the talk with a substantial
introduction for non-experts. Our result demonstrates
that...
